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Master of Science Thesis in Electrical EngineeringDepartment of Electrical Engineering, Linköping University, 2018

Impact of Engine Dynamicson Optimal EnergyManagement Strategies forHybrid Electric Vehicles

Andreas Hägglund and Moa Källgren

Master of Science Thesis in Electrical Engineering

Impact of Engine Dynamics on Optimal Energy Management Strategies forHybrid Electric Vehicles

Andreas Hägglund and Moa Källgren

LiTH-ISY-EX--18/5163--SE

Supervisor: Fatemeh Mohseniisy, Linköpings universitet

Martin SivertssonVolvo Car Corporation

Markus GrahnVolvo Car Corporation

Dhinesh VelmuruganVolvo Car Corporation

Examiner: Lars Erikssonisy, Linköpings universitet

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2018 Andreas Hägglund and Moa Källgren

Abstract

In recent years, rules and regulations regarding fuel consumption of vehicles andthe amount of emissions produced by them are becoming stricter. This has ledthe automotive industry to develop more advanced solutions to propel vehicles tomeet the legal requirements. The Hybrid Electric Vehicle is one of the solutionsthat is becoming more popular in the automotive industry. It consists of an elec-trical driveline combined with a conventional powertrain, propelled by either adiesel or petrol engine. Two power sources create the possibility to choose whenand how to use the power sources to propel the vehicle. The strategy that decideshow this is done is referred to as an energy management strategy. Today mostenergy management strategies only try to reduce fuel consumption using modelsthat describe the steady state behaviour of the engine. In other words, no reduc-tion of emissions is achieved and all transient behaviour is considered negligible.

In this thesis, an energy management strategy incorporating engine dynamicsto reduce fuel consumption and nitrogen oxide emissions have been designed.First, the models that describe how fuel consumption and nitrogen oxide emis-sions behave during transient engine operation are developed. Then, an energymanagement strategy is developed consisting of a model predictive controllerthat combines the equivalent consumption minimization strategy and convex op-timization. Results indicate that by considering engine dynamics in the energymanagement strategy, both fuel consumption and nitrogen oxide emissions canbe reduced. Furthermore, it is also shown that the major reduction in fuel con-sumption and nitrogen oxide emissions is achieved for short prediction horizons.

iii

Acknowledgments

We would first like to thank our thesis advisor Fatemeh Mohseni from the depart-ment of electrical engineering at Linköping University for all the valuable inputson the thesis.

We would also like to thank our supervisors; Martin Sivertsson, Markus Grahn,and Dhinesh Velmurugan at Volvo Cars Corporation for your commitment of be-ing our supervisors. We are grateful for your engagement in the project and forall valuable input you have provided as well as all the interesting discussions. Athank you should also be dedicated to Christoffer Strömberg at Volvo Cars Cor-poration, thank you for your valuable input about optimization.

We would also like to thank our opponents, Simon Berntsson and Mattias An-dreasson, who has given their advice about optimization along the project.

Finally we would like to thank our examiner Lars Eriksson for his expertise andenthusiasm in the subject that have had a considerable positive impact on ourstudies.

Linköping, June 2018Andreas Hägglund and Moa Källgren

v

Contents

Notation ix

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Optimization strategies . . . . . . . . . . . . . . . . . . . . . 41.3.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Risks and Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Thesis goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 The Hybrid Electric Vehicle 112.1 Series Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Parallel hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Combined Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Optimization 153.1 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Real-time optimization . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Definition of convexity . . . . . . . . . . . . . . . . . . . . . 183.3.2 Embedded Conic Solver . . . . . . . . . . . . . . . . . . . . 193.3.3 Second-order cone programming . . . . . . . . . . . . . . . 20

3.4 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Method 214.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Drive Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.1 Battery Model . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.2 Integrated Starter Generator . . . . . . . . . . . . . . . . . . 24

vii

viii Contents

4.3.3 Internal Combustion Engine . . . . . . . . . . . . . . . . . . 244.3.4 Convex Models . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4.1 Global Optimization . . . . . . . . . . . . . . . . . . . . . . 344.4.2 Real-Time Optimization . . . . . . . . . . . . . . . . . . . . 374.4.3 Embedded Conic solver . . . . . . . . . . . . . . . . . . . . 38

5 Validation 415.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1.2 Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1.3 NOx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Results 456.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1.1 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.1.2 Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.1.3 NOx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Analyses of Result 637.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 Conclusions & Future Work 678.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A Drive Cycles 73

B Tables 75

Bibliography 77

Notation

General notations

Variable Representing

P PowerT TorqueF Forcev Velocityω Rotational Speeda Accelerationθ Anglet Time

Index notations

Index Representing

f FuelNOx Nitrogen oxidesmeas Measured valuesstat Values from quasi static measurementsact Actual value at current time stepreq Requested value at current time step

trans, start Start of transienttrans, end End of transient

whl Wheelech Electrochemicalpt Power Trainquad Quadratic

ix

x Notation

Battery notations


ξ State of chargeUoc Open-circuit voltageI CurrentRi Battery internal resistanceQ Battery capacityQ0 Battery nominal capacity

Optimization notations


J Cost functionλNOx Equivalence factor for NOxλech Equivalence factor for SoCdt Sample Time

Constants

Notation Representing

ANOx Inclination of the dynamic NOx modelQLHV Lower heating value of combustionaSoC The inclination for SoC relationshipbSoC The offset for SoC relationshipα Scaling factor for variablesβ Scaling factor for equationsγ Speed ratio between ICE and ISGa1...n The inclination of the piece-wise linearized modelsb1...n The offset of the piece-wise linearized modelsak Inclination of dynamic fuel model

Aquad,NOx First term of quadratic dynamic NOx modelBquad,NOx Second term of quadratic dynamic NOx model

Notation xi

Abbreviations

Abbreviation Complete form

hev Hybrid Electric VehicleICE Internal Combustion EngineEMS Energy Management StrategyEM Electric Machine

ECMS Equivalent Consumption Minimization StrategyFTP75 EPA Federal Test ProcedureSoC State of ChargeDDP Deterministic Dynamic ProgrammingPMP Pontryagin´s Minimum PrincipleSDP Stochastic Dynamic ProgrammingMPC Model Predictive ControlEGR Exhaust Gas RecirculationVGT Variable-Geometry TurbochargerWLTC Worldwide Harmonized Light vehicles Test CyclesISG Integrated Starter Generator

RMSE Root Mean Square ErrorQCML Quadratic Cone Modeling ToolboxECU Electronic Control Unit

1Introduction

1.1 Background

In the last decade, human actions have led to dramatic environmental changesthat have devastating consequences on the environment. A rapid increase ofgreen house gases have caused higher temperatures, more extreme weather con-ditions, rising ocean levels and an increase in air pollution and will continue todo so if no arrangements are made to prevent this. The global population have be-come more aware of this issue and in response to this awareness, legislation is be-ing passed across the world to ensure these consequences are not irreversible. Amajor part of this legislation has affected the automotive industry and forced it toadapt and to primarily reduce vehicle emissions and fuel consumption. Recentlyit has also become clear that the drive cycles used for certifying this legislationdoes not capture real driving conditions. This has enabled the car industry to op-timize their vehicles to pass these simplified drive cycles while not performing aswell during real driving conditions. Therefore, tougher driving cycles that cap-ture real driving conditions, both steady-state and transient driving behaviour,are being designed and implemented.

1.2 Problem Description

A popular solution to meet the legislation passed is the Hybrid Electric Vehicle(HEV). One kind of HEV is a car that has an Internal Combustion Engine (ICE)and an Electric Machine (EM). Since the power can be provided from two dif-ferent actuators it creates the possibility to optimize how and when to engagethem, often to ensure low fuel consumption as well as low emissions. To passthe new tests, more complex aftertreatment systems are being designed to ver-

1

2 1 Introduction

ify legislation concerning emissions. On the other hand, hybridization enablesminimization of both emissions and fuel consumption at the same time if goodcontrol systems are available. Recent studies, see [1–3], has also shown that en-capsulating the dominating dynamics of the powertrain in the powertrain modelcould result in even lower fuel consumption and emissions. The HEV is increas-ingly becoming popular because it does not only pass the tougher driving cycles,it also performs better in real life.

To be able to utilize the full potential of HEVs, it is necessary to look at the EnergyManagement Strategy (EMS). The EMS developed in this thesis determines howthe torque required by the driver should be split between the two energy sourcesin order to ensure low fuel consumption and low emissions while maximizingpower utilization. The EMS can be formulated in very different ways dependingon the requirements of a specific application [4].

In this master thesis, an optimal energy management strategy will be constructedfor a mild parallel HEV that is charge sustaining. Charge sustaining means thatthe battery level should be the same at the end of the drive cycle as it was at thebeginning. The long term goal of the method developed in this thesis is to beimplemented as a real time optimal controller. A real time optimization methodoptimizes the given problem in real time which therefore, puts constraints on thecomplexity of the problem as well as the computational time of the optimizationtechnique. A common strategy to meet these requirements is to view the problemas a series of stationary operating points. However, this does neglect the transi-tion cost from one operating point to another. Therefore, models accounting foractuator dynamics will be designed and implemented. Then, an optimizationstrategy with low computational complexity and short term prediction horizonwill be designed to minimize the fuel consumption as well as the amount of nitro-gen oxides, NOx, emissions before the after treatment system. The final EMS willbe implemented and compared to optimization strategies using only static mod-els. The length of the prediction horizon will be analyzed to see how it affects theresults.

The studied vehicle is a mild parallel hybrid which is based on an ICE that runson diesel with an additional electric path. In parallel hybrids, both the combus-tion engine and the electric machine can supply the desired power, alone or incombination, which makes it possible to optimize the EMS between the two par-allel paths [5]. Figure 1.1 illustrates the studied parallel HEV powertrain config-uration. The Integrated Starter Generator (ISG) acts as an electric machine.

1.2 Problem Description 3

FD TCGB

ISG BATT

ICE FT

Figure 1.1: An illustration of a parallel HEV which contains the componentsfinal drive (FD), a gearbox (GB), a torque coupler (TC), an internal combus-tion engine (ICE), a fuel tank (FT), an integrated starter generator (ISG), anda battery (BATT). The darker rectangles represents the wheels of the vehicle.

4 1 Introduction

1.3 Literature Review

This section presents a short review of recent research studies on the topic of thisthesis.

1.3.1 Optimization strategies

There are several different optimization-based EMS and a common goal for allof them is to minimize some predefined state variables and the most commonone is fuel consumption. This is done by minimizing an objective function thatdepend on these variables. The main control optimization based strategies arerepresented in Figure 1.2. Rule-based control strategies are used for controllingfundamental control schemes, and optimization-based control strategies mini-mize an objective function [4]. The optimization-based control techniques can befurther divided in to real-time and global optimization methods.

Figure 1.2: Overview of HEV control strategies.

The global optimization strategies have the advantage of finding the global opti-mum by optimizing the complete powertrain system, given complete knowledgeof a drive cycle. Two common techniques that are used for this purpose are lin-ear programming and dynamic programming. The reader is referred to [4] formore details about these two techniques. The downside with these techniquesare that they are computationally heavy and are not suitable for real-time appli-cations. However, they are useful for validating real-time optimization strategies.

Real-time optimization methods reduce the size of the optimization problem byintroducing an instantaneous objective function that depend only on the presentstate variables. Then, a local optimum is calculated instantaneously at each timestep during a driving mission. Most of the real-time optimization strategies andthe local optimum calculations do not necessarily give a global optimum butthey often give a solution close to the global optimum. Some of the commontechniques that have been used in literature are Pontryagin’s Minimum Principle(PMP) [6, 7], Equivalent Consumption Minimization Strategy (ECMS) [8, 9], andModel Predictive Control (MPC) [3]. In both [8, 9], an ECMS has been applied

1.3 Literature Review 5

on a parallel hybrid and the results show that both fuel and NOx emissions arereduced compared to other strategies. The results in [6, 7] show that PMP is agood candidate for solving a real-time optimization problem.

MPC is a suitable method for controlling dynamic models. By taking future timeinto account, MPC optimizes the current timeslot [4]. In [3] an MPC that consid-ers the effects of the diesel-engine transient characteristics is evaluated. Thesecharacteristics become more obvious in HEV applications as there are frequenttransient operations. Since MPC takes future driving characteristics into account,it could potentially decrease fuel consumption and emissions by incorporatingthis when calculating the optimal control signals. Therefore, an MPC could havea greater impact when it is applied on a drive cycle with more transient drivingbehaviour such as rapid accelerations, etc.

Another strategy that has become increasingly popular for optimization of power-trains in HEVs is convex optimization. This is due to its computational efficiencyas well as the guarantee of finding a global optimum for a given problem. But, theoptimization problem sometimes has discrete decision variables which cannot beoptimized by convex optimization. Therefore, a good approach is to use Deter-ministic Dynamic Programming (DDP) for the discrete variables (engine on/offand gearshifts) and convex optimization to determine the optimal power split.By adding costs for switching the engine off/on and for gearshifts, it prevents theengine from doing unacceptably frequent starts and gear shifts [10]. When thismethod was compared to a basic DP algorithm, the method resulted in a reduc-tion of evaluation time and a higher precision because the convex optimizationdoes not require a discretization of the state variables and the continuous control.Another downside with convex optimization is that it requires convex modelswhich is not always possible.

1.3.2 Modeling

Research presented in [2, 3, 11–13] has shown that the main difference betweensteady-state engine operation and transient operation, with respect to emissionsand fuel consumption is caused by dynamics in the air system. Since most dieselengines are equipped with a turbo system, it is the inertia of the compressor,turbine, and turbine shaft that cause the dynamic behaviour. Therefore, it isimportant to consider them when implementing an EMS that considers transientbehaviour [12]. Results show that the optimal trajectories differ substantially andthat neglecting the turbocharger dynamics can underestimate the consumptionby over 60 %. Also the required energy needed to go to the optimal operationpoints differs from the case in which the dynamics are neglected [11].

If Variable-Geometry Turbocharger (VGT) and Exhaust Gas Recirculation (EGR)are parts to be considered, the EMS model will probably be easily modeled in twoparts. The first part calculates the injected fuel, setpoints for boost pressure, oxy-

6 1 Introduction

gen fraction in the intake manifold, and injection timing. Then, the second partconsiders the VGT and EGR. This is an approach that was used before with goodresults, see [14]. According to [14], by using an offline based transient EMS on adiesel engine, reduction of fuel consumption and the emission peaks comparedto steady-state EMS are achieved for the New European Driving Cycle (NEDC).

NOx is strongly correlated to high temperatures in the cylinder which in turn de-pends on oxygen concentration and combustion duration. A change of load leadsto increased fuelling which in turn makes the control system starve the EGR. Con-secutively, this leads to increased NOx emissions as the engine is moving towardthe desired working point [13].

Static NOx and fuel models can be acquired from static engine measurementswhere engine speed and torque are changed in a systematic order [13]. NOxemissions are however very correlated to transient effects. This is because theyare very dependent on the temperature in the cylinder which in turn depend onoxygen concentration. During a transient operation, either the engine speed orthe torque is changed which results in disturbances in the combustion chamberand air entrapment until steady-state is attained. This behaviour should prefer-ably be captured by the transient models and could be well-described by mod-elling the turbocharger lag which greatly affects the intake manifold pressure.Therefore, a dynamic model of the intake manifold pressure could be enoughto encapsulate turbocharger dynamics. A possible transient NOx model is pre-sented in [2] where the transient part of the model is modeled as a step in engineeffect multiplied with a correction factor that depend on the relative cumulativeemission mass flow errors. Results indicate that significantly lower emissions areachieved when using the model described with Equation 2 in [2].

The fuel flow can be modeled from the wheel speed and is approximated in [5]as a function of engine friction pressure, engine speed, torque, cylinder volume,lower heating value, Willans efficiency, and time.

1.4 Approach 7

1.4 Approach

The work consists of three major parts:

1. Modelling

2. Optimization

3. Analysis

In the modelling part, the models that describe the fuel consumption and theamount of NOx emissions are designed. Two sets of models are developed, staticand dynamic. The static models capture only steady-state driving behaviour andthe dynamic models capture both the behaviour during steady-state and tran-sient driving conditions. The models are designed based on data used in [13].

In the optimization part, one convex optimization tool is chosen. When a convexoptimization strategy is used, it requires the models to be convex. If the designedmodels are not convex they have to be approximated as convex functions. Otherpossible optimization strategies are for example non-convex optimization meth-ods and linearization around each working point. These methods have not beeninvestigated in this thesis, instead convex optimization is used because of its ad-vantages mentioned in section 1.3. Further explanation about convex optimiza-tion is found in chapter 3.

Finally, global optimization based EMS and real-time optimization based EMSare designed based on the models created. The real-time optimization is vali-dated against the global optimization, with and without the dynamic models andthe results are analyzed. To be able to compare the results for the different meth-ods on an even scale of performance the energy management strategies developedare charge sustaining.

8 1 Introduction

1.5 Risks and Delimitations

One of the goals for this thesis is to investigate the impact of the length of a shorttime prediction horizon on the optimal torque split and thus the fuel consump-tion and the NOx emissions. To do so, a given driving cycle will be used whichmeans that the velocity profile of the car will be known and therefore, no pre-diction is actually made. But, this is still a fair delimitation since the goal is toinvestigate if a potential velocity prediction could yield a better optimization. Ifthe optimization is not improved, trying to predict the velocity to use in an MPCis meaningless.

The developed models are based on data from engine test rigs. How this datawas produced is crucial since engine tests are done to produce data that fits acertain application. The data that is used in this thesis was developed for anotherapplication with a similar goal, though with a different approach, where a tran-sient NOx model was developed, see [13]. The transient behaviour of this modeldepend on several variables that were adjusted during the tests. However, themodel in this thesis does not depend on these variables and therefore, it might bedifficult to extract sufficient information from the given data.

1.6 Thesis goals

This thesis aims to evaluate the impact of engine dynamics on the NOx emissionsand fuel consumption. Steady-state models as well as dynamics models for thesevariables of interest will be designed. Then, they will be integrated with an HEVmodel and a global EMS as well as a real-time EMS will be developed with thegoal to minimize fuel consumption and NOx emissions.

The following questions should be answered:

• Is it possible to save fuel and reduce NOx emissions by considering dynamicactuator behaviour when developing an optimal EMS for a charge sustain-ing HEV?

• By using a prediction horizon, is it possible to save fuel and reduce NOxemissions, and how does the length of the prediction horizon affect theemissions and fuel consumption?

1.7 Outline 9

1.7 Outline

The rest of the report is organized in the following chapters.

2. The Hybrid Electric Vehicle - Facts about the HEV and its basic theory

3. Optimization Strategy - What strategies is used and theories behind them

4. Method - How the models, static and dynamic, are developed as well ashow the optimization problem is defined

5. Validation - Explanation of how the result is developed

6. Result - Presentation of the obtained result

7. Analyses - Contains analyses of result

8. Conclusions and Future Work - Conclusions are given with a discussionand some suggestions about Future Work

2The Hybrid Electric Vehicle

To improve performance, lower both fuel consumption and emitted emissions,the Hybrid Electric Vehicle (HEV) is a good alternative to the common combus-tion engine. The advantages of the HEV are the possibility to downsize the en-gine, recover some energy during deceleration, optimize the power distribution,eliminate the idle fuel consumption by turning off the combustion engine, andeliminate the clutch losses.

HEVs have two or more prime movers and power sources. In general, an HEVincludes an combustion engine as a fuel converter or irreversible prime mover.

An HEV can have different architecture designs; series, parallel, or combined hy-brid, where the most common one is the parallel hybrid with a gasoline engine.This thesis will consider a parallel hybrid with a diesel engine, where both primemovers operate on the same drive shaft. Thus, they can power the vehicle indi-vidually or simultaneously.

11

12 2 The Hybrid Electric Vehicle

2.1 Series Hybrid

The series hybrid can be seen as an electric vehicle with an additional ICE-basedenergy path since it is the electric machine that is coupled to the drive shaft. Thecombustion engine output is converted into electricity that can either directlyfeed the electric machine or charge the battery and the link between the com-bustion engine path and the battery is electrical. How the power is distributedthrough the driveline is determined by the power link which is regulated by thepower split controller. Figure 2.1 illustrates the design of a series hybrid.

The advantage of a series hybrid is that the ICE is decoupled from the drive shaftand can be operated with optimal efficiency. There is also no need of a compli-cated multi-speed transmission or clutch because the engine is decoupled andthat the EM does not need them. The disadvantage is that it requires three ma-chines which add some weight and cost to the vehicle. The overall efficiency ofusing a series hybrid will approximately be the same as for vehicles with modernICEs.[5]

FD PLEM

BATT

FT

GEN ICE

Figure 2.1: A configuration of a series HEV, which contains the parts finaldrive (FD), an electric machine (EM), a power link (PL), an internal combus-tion engine (ICE), a fuel tank (FT), a generator (GEN), and a battery (BATT).The darker rectangles represents the wheels of the vehicle.

2.2 Parallel hybrid 13

2.2 Parallel hybrid

The parallel hybrid may be considered as a conventional vehicle with an addi-tional electric path. In the parallel hybrid, both prime movers operate on thesame drive shaft which make it possible to use the electric and the fuel powerindividually or simultaneously. This makes it possible to turn the engine on/offand the electric machine can be used to assist during accelerations. The torquecoupler distributes the power flow between the actuators and is regulated in anoptimal manner by a regulator.

Since only two components are needed, there are weight and cost advantagescompared to series hybrids. However there is need for a transmission due to thefact that the ICE is mechanically coupled to the drive shaft, which adds losses tothe configuration. Figure 2.2 illustrate the components and schematic picture ofthe power train of a parallel HEV [5].

There are different ways of positioning the electric machine with respect to thetraditional drive train; micro hybrids, pre-transmission parallel hybrid, single-shaft hybrid, post-transmission parallel hybrid, double-shaft parallel hybrid, trough-the-road parallel hybrid, and double-drive parallel hybrid. For more informationabout these, see [5].

The overall efficiency of a parallel hybrid vehicle will be better than that of amodern ICE based vehicle because of brake energy recuperation and low loadelectrical operation.

FD TCGB

EM BATT

ICE FT

Figure 2.2: An illustration of a parallel HEV which contains the componentsfinal drive (FD), a gearbox (GB), a torque coupler (TC), an internal combus-tion engine (ICE), a fuel tank (FT), an electric machine (EM), and a battery(BATT). The darker rectangles represents the wheels of the vehicle.

14 2 The Hybrid Electric Vehicle

2.3 Combined Hybrid

The combined hybrid is most often a parallel hybrid which contains some fea-tures from the series hybrid. It uses both a mechanical and an electric link be-tween the engine path and the electric path and has two electric machines inaddition to the combustion engine. One of the electric machines is used as aprime mover or for regenerative braking similar to a parallel HEV. The other elec-tric machine acts like a generator, as for the series hybrid, and is used to chargethe battery via the engine or for the stop-start operation [5]. Figure 2.3 shows thedesign of a combined HEV.

FD PSDGB

BATTEM

GEN

ICE FT

Figure 2.3: The combined HEV contains the parts final drive (FD), a gearbox(GB), a power split device (PSD), an electric machine (EM), a battery (BATT),a generator (GEN), an combustion engine (ICE), and a fuel tank (FT). Thedarker rectangles represents the wheels of the vehicle.

3Optimization

For every HEV, a good EMS which decides how and when the two actuators (theICE and the EM) should be engaged is necessary to achieve good fuel economy.A good way of doing this is by using optimization techniques. Depending onthe application, that is if the EMS is to be implemented in a real-time controlleror not, the requirements and available information differ from an EMS utilizingglobal optimization.

As for all optimization methods, it is important to define the optimization prob-lem correct. The optimization problem will consist of an objective function, J(x),which states what is to be maximized or minimized. A set of constraints are alsodefined that confines the problem, see Equation 3.1

minarg x

J(x)

g(x) ≤ 0(3.1)

For an optimization problem there exist a dual problem and a primal problem,an illustration is made in Equation 3.2. If the primal problem is formulated as aminimization problem; then the dual problem is formulated as a maximizationproblem. [15] The optimization variables in the primal problem are referred toas primal variables (x) and for the dual problem as dual variables (y).

Primal: Dual:

minimize z = cT x maximize v = bT y

subject to Ax ≥ b subject to AT y ≤ cx ≥ 0 y ≤ 0

(3.2)

15

16 3 Optimization

The concept of duality is an important theory in optimization. By using this the-ory one can guarantee optimality when the solution to the primal problem equalsthe solution of the dual problem and that the solution satisfies all the constraints.For Equation 3.2, it means that optimality is achieved when z = v and x and yfulfill the constraints.

Another important concept derived from duality is Lagrangian duality. Lagrangianduality states that the optimization problem can be reformulated as in Equa-tion 3.3. In the new formulation, a certain constraint can be removed if theobjective function is reformulated with the Lagrangian function L(λ, x). It canbe thought of as introducing the constraint in the objective function with a cost,λ, called the Lagrangian multiplier. By choosing the variable λ properly, thispenalty in the objective function can result in a very similar behaviour as if theconstraint had been present. The Lagrangian multiplier for a certain constraintcan be calculated by examining the dual variable for that constraint.

Primal: Lagrangian relaxation:

minimize J(x) minimize L(λ, x) =J(x) +m∑i=1

λigi(x)

subject to gi(x) ≤ 0 i = 1, ..., mx ∈ X

(3.3)

3.1 Global Optimization

Global optimization techniques have the advantage of finding the global opti-mum since they use complete knowledge of the problem. The downside is thatthey usually are computationally heavy. When minimizing fuel consumptionand NOx emissions, the objective function can be formulated as in Equation 3.4,in which the constraints can be set for the complete drive cycle. In Equation 3.4λNOx represents a fuel equivalent factor which converts the amount of NOx emis-sions to equivalent fuel consumption. For a more detailed explanation of theequivalence factor see Section 3.2 or [5, 16].

minarg x

ṁf (x) + λNOx · ṁNOx(x)

xmin ≤ x ≤ xmax(3.4)

3.2 Real-time optimization 17

3.2 Real-time optimization

Real-time optimization techniques have the requirement of being computation-ally efficient. This puts constraints on the complexity of the problem which oftenresults in having to simplify the optimization problem. The ECMS method is apopular method when implementing a real-time optimal control energy manage-ment strategy and is derived from PMP. [17]

PMP provides necessary conditions for the optimal control of a dynamical system.When PMP is applied on the energy management problem for an HEV, the stateconstraints are neglected and a Hamiltonian is defined that has to be minimized,see Equation 3.5.

H(x(t), u(t), µ(t), t) = g(u(t), t) + µ(t) · f (x(t), u(t), t) (3.5)

In Equation 3.5, x(t) represents the state variables, u(t) the control signals andµ(t) an adjoint state, often used in optimal control theory. Under the assumptionthat the internal resistance and the open circuit voltage of the battery does notdepend on the state of charge, the adjoint state can be considered constant alongthe optimal trajectory. By introducing the costate, λ,

λ = −µ · QLHVUOCQ0

(3.6)

where QLHV represents the lower heating value of the fuel, UOC , the open circuitvoltage of the battery, and Q0, the battery’s nominal capacity, the Hamiltoniancan be rewritten as follows. [5]

H(t, u(t), λ) = Pf (w(t), u(t)) + λ · Pech(w(t), u(t)) (3.7)

In Equation 3.7, Pf represents the fuel power and Pech the electrochemical powerin the battery. The costate λ acts as an equivalence factor since the fuel powerand electrochemical power are not directly comparable. If λ is given a low value,then electrochemical power will be "cheaper" than fuel power resulting in deple-tion of the battery and vice verse. For a specific value of λ, the solution thatminimizes the Hamiltonian will represent a charge sustaining trajectory for thestate of charge. This is desirable when comparing different solutions.

When NOx emissions are introduced in the Hamiltonian, there will be need fora second equivalence factor. This equivalence factor will express the NOx emis-sions as an equivalent fuel consumption, just as λ did with the electrochemicalpower in the Hamiltonian stated above. [16]

18 3 Optimization

3.3 Convex Optimization

One approach of implementing either a global or real time energy managementstrategy could be by using convex optimization. A convex optimization problemcan be considered as a generalization of linear programming. The convex opti-mization problem has the advantage of always finding the global optimum and isoften computationally efficient. It can be described for a minimization problemon the following form,

minimize f0(x)

subject to fi(x) ≤ bi , i = 1, . . . , m.(3.8)

where the functions f0,. . . ,fm:Rn → R need to be convex. x = (x1, . . . , xm) is avector with the optimization variables, the function f0 is the objective function,and the functions fi : Rn → R, i = 1, . . . , m are the constraint functions with theconstant limits b1, . . . , bm. An optimal solution is obtained when the x vector hasthe smallest objective value among all vectors that satisfy the constraints.

In addition, in a convex optimization formulation, the constraints need to beconvex or affine functions because it ensures that no local minimum exists, andthe problem has only one global minimum [18].

3.3.1 Definition of convexity

The definition of a convex function is as follows. A function f : Rn → R, whereRn is a generic finite-dimensional vector-space and n is its dimension, is convexif its domain f is a convex set and for all x, y ∈ domainf , and θ with 0 ≤ θ ≤ 1,the following conditions hold.

f (θx + (1 − θ)y) ≤ θf (x) + (1 + θ)f (y). (3.9)

For a first order condition, it means that if f is differentiable, meaning that 5fexists at each value in f , then the function f is convex if and only if the domainof f is convex and

f (y) ≤ f (x) + 5f (x)T (y − x) (3.10)

holds for all x, y ∈ domainf .

If f is a second order system and is twice differentiable, the function is convex ifand only if the domain f is convex and its Hessian is positive semidefinite:

52f � 0

Where � denote a generalized inequality. For vectors, it represents component-wise inequality and for symmetric matrices, it represents matrix inequality [18].

3.3 Convex Optimization 19

3.3.2 Embedded Conic Solver

One software package that can be used for solving convex problems is EmbeddedConic Solver (ECOS), see [19]. ECOS is an interior-point solver for second-ordercone programming (SOCP) designed for embedded systems. The standard formfor the problem in ECOS is defined in Equation 3.11.

minimize cT x

subject to Ax = b

Gx + s ≺K h(3.11)

The matrix G and the vector h represents the inequality constraints, where thesymbol ≺K represent a generalized inequality with respect to the cone K as fol-lows.

Gx ≺K h⇔ s = h − Gx ∈ K

and the matrix A with the vector b represents the equality constraints. The vec-tor s represents slack variables and K the cone. x is a vector with the primalvariables and c is a vector that determines and weights which variables are to beminimized.

To avoid numerical problems, it is a good idea to scale all the primal variables tovalues within the same short range, for example the range [-1,1]. ECOS requiresthe matrices A and G to be sparse matrices. Meaning that they have to be con-verted from full matrices into sparse form. This saves memory and is done inMATLAB with the commando sparse. A function call to ECOS is made with thefollowing command:

[] = ecos(c’,G,h,dims,Aeq,beq,opts)

where dims determines how many constraints exist, opts tells ECOS what op-tions to use when solving the problem, and the rest are the matrices/vectors ex-plained above. For more information about ECOS the reader is referred to [19].

20 3 Optimization

3.3.3 Second-order cone programming

SOCP can cast problems like Matrix-fractional and Quadratically constrainedquadratic programming. A brief explanation of SOCP is that it is a problem classthat lies between linear or quadratic and semidefinite programming and it canbe solved very efficiently by using primal-dual interior-points methods [20]. Anexample of a quadratic constraint is given in Equation 3.12. The second equationis written as a second-order cone and is equivalent to the first constraint equation.

xTATAx + bT x + c ≤ 0∥∥∥∥∥(1 + bT + c)/2Ax∥∥∥∥∥

2≤ (1 − bT − c)/2

(3.12)

3.4 Model Predictive Control

The basic idea of a Model Predict Control (MPC) is to formulate the problem asan optimization problem and solve the problem on-line at each time when newmeasurement signals are obtained. An on-line optimization requires fast calcula-tion time, and therefore an MPC can be a good technique.

An MPC predicts the future trajectories by using measurements from currenttime and control signal during each prediction horizon. If the goal is to solve aminimization problem, the objective function should be minimized while all theconstraints should be satisfied. After the MPC implements the first step of thecontrol sequence it moves the prediction horizon one step forward and repeatsthe optimization procedure. This is repeated for the whole drive cycle [21].

The prediction horizon is set to a specific length before running the optimizationproblem. A common way of choosing the length of the prediction horizon is tocover a typical settling time of the desired closed system. [22].

4Method

In this chapter a detailed explanation is given on how the powertrain is modeledwith extra focus on the fuel and NOx models. In addition, an explanation abouthow the optimization problem is set up using the developed models is provided.As mentioned earlier, the aim of the optimization problem defined in this thesisis to minimize NOx emissions and fuel consumption while maximizing powerutilization.

4.1 Motivation

Since the requirements for a real-time EMS include both high accuracy and lowcomputational time, it is desirable to use convex optimization techniques. Investi-gation of the static NOx and fuel maps obtained from steady-state measurementsof the studied diesel engine shows a close-to-convex behaviour. Since the dy-namic models will be an extension of the static maps, it seems reasonable to useconvex optimization. However, if the convex models does not prove to be accu-rate enough, a different method will be used to be able to answer the questionsstated in section 1.6.

4.2 Drive Cycle

To compare the performance of different vehicles, for example the amount ofemissions and fuel consumption, and to ensure that legislation is enforced, stan-dard test cycles are used. All newly-manufactured vehicles has to meet the legalrequirements, and for different selling markets, there are different drive cyclesthat are used. The WLTC was developed to represent typical driving conditions

21

22 4 Method

around the world. It is based on driving data collected around the world (EU,India, Japan, Korea, USA) combined with suitable weight factors, see [23]. Thevelocity profile for the WLTC drive cycle is represented in Figure A.1. One drivecycle that is used in the EPA Federal Test Procedure is the FTP-75 cycle, whichwas developed to measure tailpipe emissions and fuel economy of passenger carsand mimic city driving. In this thesis, both of these diving cycles are used toevaluate and compare the NOx emissions and fuel consumption. The velocityprofile for the FTP75 drive cycle is presented in the appendix and is representedin Figure A.2. In addition to the WLTC and FTP75 drive cycles a random drivecycle that encapsulates city driving, in this report referred to as City drive cycle,is investigated and is presented in the appendix, see Figure A.3.

When applying global optimization techniques on the drive cycles mentioned inthe paragraph above with a time step small enough to capture the engine dynam-ics, the computers available ran out of physical memory. Therefore segments ofabout 1000 seconds are evaluated for each drive cycle.

4.3 Models

In order to be able to optimize how a vehicle should use its actuators, the powerrequest at the torque coupler should be calculated. For this purpose, a model ofthe powertrain, the vehicle and the speed profile is needed. In this thesis, no vehi-cle model is developed, instead data is collected using VSim. VSim is an in-houseSimulink-based simulation tool used at Volvo Cars Corporation for analysis ofthe vehicles fuel economy and performance. In VSim, a mild parallel hybrid carwith correct components is chosen along with a drive cycle. A simulation is madeand relevant data is extracted. The data that are needed for simulation are theengine speed, weng , power request at the torque coupler, Preq,pt , the engine on/offstatus, engon, and the time, t.

The optimization outputs the optimal power split ratio for the torque couplerthat is needed to meet the speed request from the driver/drive cycle. This powerneeds to be delivered by the actuators. Therefore, models for the ISG, the ICEand the battery, see Figure 2.2, need to be developed in order to set up the opti-mization problem.

For these components, static models are developed that only capture the steady-state behaviour. The static models for NOx emissions and fuel consumption arethen expanded in order to capture the transient behaviour when going from onestationary working point to another. Since the applied optimization method isconvex optimization, all of these models have to be convex.

In the remaining parts of this chapter, first the procedure of developing the mod-els for each component that are going to be optimized is presented. Second, thedynamic fuel and NOx models are presented in detail. Finally, there is a detailed

4.3 Models 23

explanation on how the convex models were developed.

4.3.1 Battery Model

The battery used in a hybrid powertrain consists of a large number of cells thatare connected in series and/or in parallel. This leads to a complex electrochem-ical model based on partial differential equations [24], and is not suitable to beused in an energy management context. Therefore, a Thevenin equation circuitis used, see [25], which is visualized in Figure 4.1. By using this model, only theState of Charge (SoC) state is dynamic. Below, SoC is represented by ξ and is theratio between the capacity of the battery (Q) and its nominal capacity (Q0), seeEquation 4.1.

Figure 4.1: Thevenin equivalent circuit model of a battery were Uoc is theopen-circuit voltage, Ri is the internal resistant, Ibatt the battery current,and Ubatt the battery voltage.

ξ(t) =Q(t)Q0

(4.1)

SoC is defined in the range ξ ∈ [0, 1]. To prohibit battery damage, which occurswhen the battery is discharged or charged to its limits, SoC is limited by an upperbound and a lower bound. The battery open circuit voltage and inner resistancedepend on the SoC. This dependency is small but still present and for the inves-tigated battery, it has a linear behavior in the range ξ ∈ [SoCmin, SoCmax], seeFigure 4.2. Therefore, the SoC is limited to the range ξ ∈ [SoCmin, SoCmax].By combining the definition of power and Ohms law the following equations areobtained, see Equation 4.2:

Pech = Uoc · IBAT TPBAT T = UBAT T · IBAT T

PBAT T ,loss = URi · I2BAT T

(4.2)

From these equations the power loss for the battery can be expressed as in Equa-tion 4.3.

24 4 Method

y

x

SoCmin SoCmax

Figure 4.2: An illustration of how the allowed values for SoC is chosen. They-axis represent the open circuit voltage of the battery and the x-axis repre-sent the SoC.

PBAT T ,loss =RiU2oc

P 2ech (4.3)

The inner resistance and open circuit voltage can be modeled as constants or asfunctions of the SoC. In this thesis Equation 4.4 is used to model both dependen-cies with one model, where the SoC is limited to ξ ∈ [SoCmin, SoCmax].

1a · ξ + b

P 2ech ≈RiU2oc

P 2ech (4.4)

4.3.2 Integrated Starter Generator

For the ISG, a static power-loss map has been developed that expresses the power-loss of the component as a function of output power and rotational speed, seeEquation 4.5. The static map only covers a set of stationary data points for acertain range in rotational speed and ISG output power. For values between thesestationary points, linear interpolation is used and for values outside the rangelinear extrapolation is used based on the inclination between the last two datapoints in the data set. The constant γ is the ratio between engine speed and thespeed of the electric machine.

PISG,loss = f (PISG, ωICE · γ) (4.5)

The dynamics of the ISG is assumed to be small enough to be neglected.

4.3.3 Internal Combustion Engine

For the combustion engine, a static and a dynamic model for NOx emissions andfuel consumption were developed. The static models capture only the steady-state behaviour whereas the dynamic models also capture the transient behaviour.

4.3 Models 25

Static Models

The static models used for both the fuel mass flow and NOx mass flow are staticmaps based on steady-state measurements done on the engine. These maps weredeveloped in [13].

FuelThe static fuel model gives a steady-state relationship between engine speed, en-gine output power and fuel mass flow, see Equation 4.6.

ṁf = f (PICE,act , ωICE) (4.6)

ṁf ·QLHV = PICE,act + PICE,loss (4.7)

PICE,loss = f (PICE,act , ωICE) (4.8)

By using Equation 4.6 and Equation 4.7, a map that describes the power-losses ofthe engine that only covers a set of stationary points is obtained, see Equation 4.8.To extract values between these points linear interpolation/extrapolation is doneas described in subsection 4.3.2.

NOxThe steady-state NOx map relates a certain NOx mass flow for a limited combina-tions of engine speeds and engine output powers using Equation 4.9. For enginespeeds and engine torques between these stationary points the same interpola-tion/extrapolation method is used as described in subsection 4.3.2.

ṁNOx = f (TICE,req, ωICE) (4.9)

Dynamic Models

The dynamic models are an extension of the static models. To ensure that the dy-namic model is convex, a dynamic part is added to the static model. If the staticmodel and the dynamic part are convex by themselves, the sum of them will alsobe convex. The dynamic part is modeled so that it captures the NOx emissions/-fuel consumption when going from one stationary point to another.

26 4 Method

FuelData from [13] is used to develop the dynamic fuel model. For the positive tran-sients, that is when going from one stationary working point to another, the dif-ference between the actual mass flow and the mass flow given by the static fuelmodel (∆ṁf ) is plotted as a function of the difference between the requestedtorque and the actual torque for different engine speeds, see Equation 4.10. Thestudy was done for 7 different engine speeds, equally distributed.

ṁf ,meas − ṁf ,stat = f (TICE,req − TICE,act) (4.10)

∆ṁ

fuel

Neng,1

∆ṁ

fuel

Neng,3

-50 0 50 100 150 200 250

Treq

- Tact

[Nm]

∆ṁ

fuel

Neng,7

Figure 4.3: Illustration of Equation 4.10. Blue crosses represent data pointsand the black line the model. Only engine speeds 1, 3 and 7 are illustrated,of the total 7 studied engine speeds.

The relationship between ∆ṁf and TICE,req − TICE,act can be approximated by alinear function for a specific engine speed, see Figure 4.3. Therefore, a simplelinear model was developed using the least square method, see Equation 4.11.The variable ak is the slope of a straight line and is a function of the engine speedωICE . ak is obtained for a specific engine speed using interpolation as explainedin subsection 4.3.2.

∆ṁf = ak(ωICE) · (TICE,req − TICE,act)ṁf ,dyn = ṁf ,stat + ∆ṁf

(4.11)

4.3 Models 27

NOxThe same approach used for the dynamic fuel model was used for the dynamicNOx model. However the NOx peaks have an offset in time to when the torquestep is made, see Figure 4.4. This offset is not constant and is probably causedby sensor dynamics and efforts of compensating for this offset. Most likely itdoes not represent the actual relationship between a transient engine operationand the resulting NOx emissions. As explained in [2, 3, 11–13] and Section 1.3.2,a transient engine operation occurs due to a change in engine speed or engineload. This in turn causes a disturbance in the combustion chamber and the airentrapment until steady-state engine operation is attained. Since NOx formationis highly dependent on the temperature in the engine cylinders which duringan engine transient will increase, it may lead to a NOx peak. Therefore, it isreasonable to assume that the delay is caused by sensor dynamics and the NOxpeak occurrs at the same time as the torque step.

ṁNOx

1113.2 1113.4 1113.6 1113.8 1114 1114.2 1114.4 1114.6 1114.8 1115

Time [s]

To

rqu

e

Figure 4.4: Illustration of the offset in time between a torque step and theNOx peak.

To find a relationship between the torque step and the additional NOx emissionsresulting from this torque step, several approaches were tested. The approachclosest to have a reasonable relationship was Equation 4.12.

∆NOx = ln( t=ttrans,end∫t=ttrans,start

NOxmeas − NOxstat0.9t

dt)

= f (TICE,req − TICE,act) (4.12)

28 4 Method

Figure 4.5 shows ∆NOx as a function of Treq − Tact defined in Equation 4.12.∆NOx

Neng,1

∆NOx

Neng,3

0 50 100 150 200Treq

-Tact

[Nm]

∆NOx

Neng,7

Figure 4.5: Illustration of Equation 4.12 where the blue crosses representsdata and the black line is the model. Only engine speeds 1,3 and 7 are illus-trated, out of the 7 studied engine speeds.

When ∆NOx was added to the static model and compared to the measured val-ues, the dynamic model (static NOx plus ∆NOx) did not behave as the measure-ments did. Therefore, a different dynamic NOx model had to be found.

Another NOx model that was evaluated was inspired by [2], see following Equa-tion 4.13.

ṁNOx,dyn = ṁNOx,stat · (1 + c ·TICE,act(tk) − TICE,act(tk−1)

Ts)

c =mNOx ,tot −

∑Nk=1 ṁNOx,stat (tk) · Ts∑N

k=1 ṁNOx ,stat(tk) · Ts ·max(TICE,act(tk )−TICE,act(tk−1)

∆t , 0)

(4.13)

In Equation 4.13, the index tot refers to the cumulative sum of measurementsmade for the complete drive cycle that the model is made for, i.e. the modelin [2] is cycle dependent. The index stat represent values interpolated from asteady-state engine map, and Ts is the sampling time.

However, the model developed in [2] is not convex and would have to be modi-fied to be used in convex optimization method. This was attempted and evaluatedwithout obtaining a good model.

4.3 Models 29

A quadratic NOx model, see Equation 4.14, was also investigated but with nosuccess. It gives positive NOx mass flows at negative transients, because the con-stant Bquad,NOx could not be tuned in a way which would compensate for thepositive contribution that is made by the first quadratic term containing the con-stant Aquad,NOx.

ṁNOx,dyn = ṁNOx,stat + Aquad,NOx ·∆T2 − Bquad,NOx ·∆T

∆T = TICE,req − TICE,act(4.14)

The model used in this thesis, see Equation 4.15, is a linear model based onthe characteristics seen in Figure 4.5 as well as it being physically reasonable.The model is fitted using the cumulative sum of the measurements from [13] bytuning the constant ANOx. The variable ∆T is the same variable used in Equa-tion 4.14.

ṁNOx,dyn = ṁNOx,stat + ANOx ·∆T (4.15)

Validation of the models used is found in chapter 5 and chapter 6.

Engine Torque

Since the purpose of this thesis is to evaluate the impact of engine dynamics, amodel that captures the major dynamics of the engine is needed. The dominat-ing dynamics for the engine is caused by the turbo lag which causes the enginetorque to lag behind the requested torque. By investigating measurement data ofthe engine torque obtained from [13], a model is developed and fitted. The torquebehaves like a first order system and is modeled using Equation 4.16 where thetime constant τ need to be determined. This is done by analyzing the characteris-tics of the torque steps.

TICE,act(t + 1) = TICE,act(t) +TICE,req(t) − TICE,act(t)

τ∆t (4.16)

4.3.4 Convex Models

In order to be able to construct a convex optimization problem, the objectivefunction and the constraints need to be convex or concave, see section 3.3. Notethat since the velocity profile of the car as well as the selected gear is consideredto be known the engine speed can be calculated. Therefore, the models for eachcomponent need only depend on the output power in a convex/concave order,depending on if something is minimized/maximized.

The dynamic extension that is added to the static models for NOx emissions andfuel consumption are convex. However, the static maps for each component ex-cept for the battery are not convex. The battery losses can be expressed as inEquation 4.17 which is a convex expression. aSoC represents the inclination andbSoC the offset for the relationship between the UOC and the SoC.

30 4 Method

PBAT T ,loss =Ri

Uoc(ξ)2P 2ech =

P 2echaSoC · ξ + bSoC

(4.17)

The static maps for the NOx mass flow, the power-losses for the ICE and thepower-losses for the ISG indicate a close to convex behaviour which is one of thereasons convex optimization was chosen. The procedure of making these staticmodels convex is done through piecewise linearization.

Piecewice Linearization

Piecewise linearization is illustrated in Figure 4.6 and it is applied on the staticmaps listed above. To ease understanding, we consider the power loss model forthe ICE but the concept is exactly the same for the other static maps. For a set ofpredefined engine speeds, the power losses are approximated with a number ofstraight lines whose slopes are increasing with increasing output power, PICE,act .By taking the maximum value of all straight lines for a specific output power, avalue close to that of the non convex model is obtained. Considering Figure 4.6,y and x can represent PICE,loss and PICE,act respectively and this would be forone specific engine speed. The number of lines for each engine speed is a designvariable and the process is repeated for a predefined number of engine speeds un-til a sufficiently correct convex map is obtained. In order to extract informationfrom the map for a engine speed that is not explicitly defined in the convex maps,the same interpolation/extrapolation method as explained in subsection 4.3.2 isused.

y

x

Figure 4.6: An illustration of piecewise linearization.

4.4 Optimization 31

4.4 Optimization

By using the models developed in section 4.3, the optimization problem is con-structed. The aim of the optimization is to minimize fuel consumption and NOxemissions while maximizing power utilization by optimizing the torque split.The optimization problem will be formulated as a global optimization problemas well as a real-time optimization problem using MPC and ECMS. These two op-timization strategies will then be divided into to subsets, one only utilizes convexstatic models and the other one uses convex dynamic models, see Figure 4.7. Theoptimal torque split for the different optimization methods and the effect that ithas on NOx emissions and fuel consumption will then be evaluated using twodifferent plants.

The two plants are referred to as Plant 1 and Plant 2. Plant 1 is in this thesisrepresented by the convex dynamic models constructed in this thesis. Plant 2 isrepresented by the non-convex static maps with the same dynamics used in Plant1, that is the convex dynamic extension for both fuel and NOx. The fuel modelused in Plant 2 is described by Equation 4.11 where ṁf ,stat is the non-convexstatic fuel map. The NOx model used in Plant 2 is represented in Equation 4.15where ṁNOx,stat refers to the non-convex static NOx map.

By analyzing the results obtained from Plant 1, an answer to the questions statedin section 1.6 is obtained under the assumption that the controller has perfectmodels describing the plant. The results obtained when using Plant 2 will in-stead answer the same questions but for the scenario when the controller doesnot have perfect models describing the plant.

Since no driver model is constructed, the modeled torque is implemented in thestatic optimization where the requested torque represents the driver and the ac-tual torque are the output from the engine. This is a reasonable simplificationthat can answer the questions in section 1.6.

In order to be able to compare the different methods, the solution obtained fromthe optimization needs to be charge sustaining. It means that the final value forthe battery SoC has to be the same (within reasonable tolerances) as the startvalue of the SoC.

The software package used for setting up the optimization problem is ECOS. Toimplement the MPC, ECOS will be used since it is suitable for a real-time con-troller.

For simplicity of notations, all variables are expressed in terms of power. Theequivalent power of a certain fuel mass flow is calculated by using Equation 4.18where QLHV is the lower heating value for diesel. The NOx mass flow equivalentpower is calculated the same way but is not a physical quantity and should bethought of as a scaled up NOx mass flow.

32 4 Method

Figure 4.7: Illustration of the controller, where the different EMS are imple-mented, and the two plants used in this thesis.

Pf = mf · qLHVPNOx = mNOx · qLHV

(4.18)

In the next section, a description is given on how the global and real-time (MPC)optimization problems are constructed followed by an explanation on how theyare implemented in ECOS.

4.4 Optimization 33

Convexity

In subsection 4.3.4, the approach that was used when the convex models weredeveloped is explained. There exist limits on the maximum and minimum powerfor the different components which are given as 1 dimensional look up tables.However, they only are dependent on the rotational speed which is given andtherefore, are a known constant in each time step. Hence, they do not need tobe convex but are still modeled using piecewise linearization and the given 1 di-mensional look up tables. For the battery, the maximum and minimum limits areconstant values that are independent of time.

Equivalence Factors

There are two equivalence factors that are used in this thesis, λNOx and λech.λNOx was obtained by one of our supervisors at Volvo Cars Corporation by calcu-lating the equivalent fuel consumed (using engine measures such as the EGR andfuel timing) for reducing NOx emissions. It weights one gram of NOx equal toone gram of fuel. λech is derived using theory briefly explained in chapter 3. It isobtained by solving the global optimization problem stated below and extractingthe dual variable correlated to the following Equation 4.19.

Pech(t) =SoC(t) − SoC(t + 1)

dt·Q0 ·UOC (4.19)

The equivalence factor λech is further explained in subsection 4.4.2.

Convex relaxation

When implementing the convex models that were created with piecewise lin-earization, a convex relaxation has to be made since the max-function does notnecessarily have a continuous first order derivative. Instead, if using the exam-ple in section 4.3.4 where the max-function is used in the same way as below, aconvex relaxation is made as in Equation 4.21.

PICE,loss = max(a1 · Pice,act + b1, a2 · Pice,act + b2, . . . , an · Pice,act + bn) (4.20)

In the rest of this section, Equation 4.20 is substituted with the convex relaxationin Equation 4.21, that can be implemented for convex optimization problems. Aslong as PICE,loss or a variable that depend on it is being minimized the modelapproximation will be valid.

PICE,loss ≥ a1 · PICE,act + b1PICE,loss ≥ a2 · PICE,act + b2

...

PICE,loss ≥ an · PICE,act + bn

(4.21)

34 4 Method

In the Equations above, n represents the number of lines used when approximat-ing a function with a piecewise linear function, and the a:s and b:s represents theinclinations and offsets of the lines. This relaxation is made for all losses, i.e. forthe battery, the ISG and the ICE as well as for the NOx emissions.

4.4.1 Global Optimization

The global optimization has the objective of determining the optimal torque splitthat minimizes fuel consumption and NOx emissions. It uses complete knowl-edge of the drive cycle and the optimization problem is formulated as below. First,the static global optimization problem is defined followed by the dynamic globaloptimization problem.

Static Optimization

The objective function for the static optimization is defined as in Equation 4.22.

[PICE,act PISG] =argmin J

J =dt · [Pf ,stat(PICE,act)]+

dt ·λNOx · [PNOx,stat(PICE,act)]

(4.22)

In Equation 4.23 and Equation 4.24 the equality and inequality constraints thatdefine the static optimization problem are represented where Uoc and Q0 repre-sent the open circuit voltage and the nominal capacity respectively. The constantsa and b for the ICE, NOx and ISG are the slopes and offsets for the straight linesconstructed when creating the convex static maps using piecewise linearization,see section 4.3.4.

Note that the sum of the produced torque from the ICE and ISG (PICE , PISG) isallowed to be greater than the requested torque (Preq). If the optimization is donecorrect, this will only occur for negative torques which cannot be supplied bythe two actuators. This means that the driver would need to apply the vehiclefriction brakes in order to achieve the requested torque.

4.4 Optimization 35

Equalities:

Pech(t) =SoC(t) − SoC(t + 1)

dt·Q0 ·UOC

Pech(t) = PBAT T ,loss + PISG,act + PISG,loss + Paux

PICE(t + 1) = [PICE(t) +PICE,req(t) − PICE(t)

τ· dt] ·

ωICE(t + 1)ωICE(t)

SoC(t = 1) = SoCstart

(4.23)

Inequalities:

ISG equations:

PISG,loss(t) ≥ 0PISG(t) ≥ PISG,min(t)PISG(t) ≤ PISG,max(t)PISG,loss(t) ≥ aISG(ωISG) · PISG(t) + bISG(ωISG)

ICE equations:

PICE,loss(t) ≥ aICE(ωICE) · PICE,act(t) + bICE(ωICE)PICE,act(t) ≥ PICE,min(t)PICE,act(t) ≤ PICE,max(t)Pf (t) ≥ 0PICE,loss(t) ≥ 0Pf (t) ≥ PICE,act(t) + PICE,loss(t)Preq(t) ≤ PICE,act(t) + PISG(t)

NOx equations:

PNOx(t) ≥ aNOx(ωICE) · PICE(t) · qLHV + bNOx(ωICE) · qLHVPNOx ≥ 0

Battery equations:

PBAT T ,max ≥ PISG(t) + PISG,loss(t) + PauxPBAT T ,min ≤ PISG(t) + PISG,loss(t) + PauxSoC(t) ≤ SoCmaxSoC(t) ≥ SoCminSoC(t = tend) ≥ SoCstart

PBAT T ,loss ≥P 2ech

aSoC · SoC(t) + bSoC(4.24)

36 4 Method

Dynamic Optimization

The objective function for the dynamic global optimization is defined in Equa-tion 4.25.

[PICE,req PISG] = argmin J

J = dt · [Pf ,stat(PICE,act) + Pf ,dyn(PICE,act , PICE,req)]

+ dt ·λNOx · [PNOx,stat(PICE,act)

+ PNOx,dyn(PICE,act , PICE,req)]

(4.25)

The indexes req and act refer to the requested power and the actual output powerof the actuator respectively. These powers will be different for the ICE due to thedynamics, but for the ISG the power will be equal since the dynamics are ne-glected in the optimization.

Equation 4.26 and Equation 4.27 are added to the static problem defined by Equa-tion 4.23 and Equation 4.24 to reflect the dynamics of the system. The constrainton the power of the fuel, Pf , in Equation 4.24 (Pf (t) ≥ PICE,act(t) + PICE,loss(t))is replaced by: Pf (t) ≥ PICE,act(t) + PICE,loss(t) + PICE,dyn(t), represented in Equa-tion 4.27.

Equalities:

∆T (t) =PICE,req(t) − PICE,act(t)

ωICE(t)(4.26)

Inequalities:

PICE,dyn(t) ≥ak(ωICE) · (PICE,req(t) − PICE,act(t))

ωICE(t)

PICE,dyn(t) ≥ 0Pf (t) ≥ PICE,act(t) + PICE,loss(t) + PICE,dyn(t)

PNOx,dyn(t) ≥ ANOx ·∆T (t) · qLHVPNOx,dyn(t) ≥ 0

(4.27)

The factor τ is the time constant for the torque dynamics of the ICE and ak isa speed dependent inclination for the linear model capturing the extra fuel con-sumption due to transient engine operation.

4.4 Optimization 37

4.4.2 Real-Time Optimization

The purpose of the real-time optimization strategy is to find the optimal torquesplit that minimizes both fuel consumption and NOx emissions. Unlike theglobal optimization strategy, the MPC in this thesis does not utilize completeknowledge of the drive cycle. Instead it has limited look ahead knowledge de-fined by a predefined prediction horizon. The solutions obtained from the MPCand the global optimization are set to be charge sustaining in order to to make afair comparison between the different methods. The ECMS approach is appliedand a Hamiltonian is introduced and minimized by finding the optimal torquesplit. The Hamiltonian is defined in a different way for the static and dynamicoptimization. An illustration of the MPC is represented in Figure 4.8.

Figure 4.8: Flowchart for the MPC. The subproblem M defines the problemthat the MPC solves for each iteration.

38 4 Method

Static Optimization

The Hamiltonian that is to be minimized using the static MPC is defined as fol-lows.

H =k∑i=1

Pf ,stat(PiICE,act) + λNOx · PNOx,stat(P

iICE,act) + λech · Pech(P

iISG, SoC

i)

(4.28)where k is the prediction horizon. The optimization problem for the static MPCis then defined in Equation 4.29.

[PICE,act PISG,act] = argmin dt ·H (4.29)

The same equations stated in Equation 4.23 and Equation 4.24 are used except forthe constraint SoC(t = end) ≥ SoCstart . The equivalence factor λech multipliedwith Pech is instead added to the cost function. By tuning λech correctly, a chargesustaining trajectory is obtained.

Dynamic Optimization

The Hamiltonian that is to be minimized in the dynamic optimization is repre-sented in Equation 4.30 and the optimization problem for the dynamic MPC isdefined in Equation 4.31.

H =k∑i=1

Pf ,stat(PiICE,act) + Pf ,dyn(P

iICE,act , P

iICE,req)

+ λNOx · [PNOx,stat(PiICE,act) + PNOx,dyn(P

iICE,act , P

iICE,req)]

+ λech · Pech(PiISG, SoC

i)

(4.30)

[PICE,req PICE,act] = argmin dt ·H (4.31)

The equalities and inequalities stated in Equation 4.26 and Equation 4.27 areused for the MPC as well except for the terminal constraint on the SoC, that isSoC(t = end) ≥ SoCstart . Similar to the static MPC, the term λech · Pech is insteadadded to the cost function.

4.4.3 Embedded Conic solver

ECOS uses matrices to solve the optimization problem in which all the constraintsare formulated in matrix form. The optimization is time dependent where differ-ent equations are needed for different time steps. Therefore, the matrices G andA are constructed for every time step and then placed in separate higher dimen-sional matrices. They are placed in the diagonal of the new matrices. This makesit possible to use previous values in an efficient way by having the smaller matri-ces for different time steps overlap each other in the higher dimensional matrix

4.4 Optimization 39

Figure 4.9: Illustration of how the matrices are placed diagonal after eachother for every time step, in order to create one higher dimensional matrix.The lighter gray represents zeros and each blue rectangle represents a matrixfor one time step. Here n time steps are assumed. The equality is usedfor A and b, see Equation 4.32, and the inequality is used for G and h, seeEquation 4.33.

as illustrated in Figure 4.9. Parameters b, h, and c are row vectors and not matri-ces so they are placed after each other for every time step. c is not illustrated inFigure 4.9 but is the vector defining the cost vector and the same logic applies forc as for b and h.

Second order cone

The quadratic constraint in Equation 4.24 has to be implemented in ECOS as asecond-order cone, see Equation 3.12. The conversion from a quadratic constraintto a SOCP is done by using the python toolbox Quadratic Cone Modeling Toolbox(QCML), see [26].

Scaling

As mentioned in subsection 3.3.2, scaling factors are needed in order to avoidnumerical problems. A specific scaling factor, α, is derived for every variableand a specific scaling factor β is derived for every equation. The scaling factorα scales the variables to be in the same value range and as a result numericalproblems can be avoided. The variable β scales the equations and is used tomake the solver find a solution faster by prioritizing certain equations. For abetter understanding of how the scaling is made, see Equations (4.32) – (4.34).

Aαβx =

bβ

(4.32)Gαβx ≤ h

β(4.33)

cT ·α (4.34)

5Validation

The validation is divided into two main parts. The first part is model valida-tion, which is important to answer the first question posed in this thesis, seesection 1.6.

– Is it possible to save fuel and reduce NOx emissions by considering dynamicactuator behaviour when developing an optimal EMS for a charge sustainingHEV?

To be able to answer the question stated above, a comparison is made on how thesolution obtained from the controller using the dynamic models differs from thesolution obtained from the controller using only the static models. This compari-son is made for both the global optimization and the MPC using both Plant 1 andPlant 2.

To answer the second question posed in this thesis,

– By using a prediction horizon, is it possible to save fuel and reduce NOx emis-sions, and how does the length of the prediction horizon affect the emissions andfuel consumption?

the solutions obtained from the MPC with different prediction horizons are in-vestigated to find the most suitable prediction horizon for the static model andthe dynamic model. To isolate the effect of engine dynamics, an iterative bisec-tion algorithm is developed that find the equivalence factor, λech, that results in acharge sustaining optimal solution. The algorithm is applied on each drive cyclefor a set of different prediction horizons. All three drive cycles are investigatedfor both static and dynamic models.

41

42 5 Validation

5.1 Models

The validation variables that have been used here are RMSE, error of variance,and errors of cumulative sums. The RMSE is the mean of the relative error be-tween the evaluated model and the measured value at each sample time.

Since there is a varying offset in time between the torque step and the correspond-ing NOx peak, see Figure 4.4, the RMSE cannot be used since there is no way toknow where the NOx peak should be if no sensor delays, compensation of thesedelays, etc would have been present. Therefore, an error of variance is used toget measurable values on how well the models capture transient behavior. This isdone by taking the difference between the model and a reference signal that hasthe shape of a square wave. The reference signal represents a completely staticbehaviour with instantaneous change in value when going from one static pointto another one. Then, the variance is calculated for the difference between themodel and the reference signal. This value gives an indication on how well amodel captures the transient behaviour when compared to the variance for thedifference between the actual values (measurements) and the reference signal.The error of variance is then expressed as the relative error between the variancefor the reference signal subtracted from the model and the reference signal sub-tracted from the measured values, where the latter represents the correct valueof the variance.

The error of cumulative sum is also expressed as a relative error, where the cumu-lative sum of the measured signal represents the correct value.The developed models for fuel and NOx are validated using data from measure-ments produced in [13] for both Plant 1 and 2. In Table 6.2 and Table 6.6, Plant1 is referred to as the Convex Model and Plant 2 is referred to as the Non-ConvexStatic Model.

When validating the different optimization methods, the two different plants areused as illustrated in Figure 4.8 and explained in section 4.4.

5.1.1 Torque

In order to get an accurate and exact validation, all models are validated with themodeled torque; therefore, the torque is validated first. As mentioned before thetorque model is a first order system fitted to measurement data produced in [13],and therefore, it is validated against the measured data.

Validation of the torque model is done using RMSE and the error between thecumulative sums, and also, with a figure to visualize the comparison between themodeled and measured torques. The cumulative sum is always calculated usingthe same length of the same drive cycle.

5.2 Optimization 43

5.1.2 Fuel

Since the convex static fuel models are an approximation of static models, theyare validated against each other. But also, in order to get a better understand-ing of how the convex static fuel maps match reality, they are validated againstmeasurement data [13]. An illustration for two torque steps is made with all themodels to visualize how they relate to each other.

5.1.3 NOx

The same validation method as for fuel is used for NOx, and the convex staticmaps are validated against both static maps and measured data. The errors arerepresented in Tables 6.6 and 6.7 and the different models are illustrated in Fig-ures 6.3 to 6.6 that are presented in chapter 6.

5.2 Optimization

In this section, the results obtained from the different optimization techniquesare presented. First, the torque trajectories for the ISG and ICE are shown for thesolution obtained from both the global optimization and the MPC, using bothstatic and dynamic models, in order to emphasize the difference between the so-lutions when considering engine dynamics. Thereafter, the cost function is eval-uated for each drive cycle and several prediction horizons including the globalsolution using both the static and dynamic models. These results are presentedboth as figures in chapter 6 and in a table in Appendix B.

The SoC trajectory is also investigated for each drive cycle, both for the static andthe dynamic controller, and are presented in in figures in chapter 6.

Another variable of interest is the calculation time for the MPC and how it changeswhen using different prediction horizons. Therefore, an illustration of the corre-lation between the length of the prediction horizon and the computational timeis made in figures presented in chapter 6.

6Results

In this chapter, all the results are presented and for more explanation of what ispresented read associated sections in chapter 5. The sections in this chapter arethe same as in chapter 5 to simplify for the reader how a specific result was ob-tained. The developed models are presented first and thereafter the optimizationmethod will be described.

6.1 Models

In this section, the results for the torque, fuel, and NOx models are presented.

6.1.1 Torque

For the torque model, the calculated error between the model and measurementsare presented in Table 6.1. How the model fits the measured torque is illustratedin Figure 6.1. The Cumulative sum error is very small and the RMSE is relativelysmall. The relative error becomes very large for engine torques close to and belowzero. For larger absolute values of the engine torque the relative error is a lotsmaller.

Table 6.1: Validation of the torque model, using measured torque values.

Torque RMSE [%]Modeled Torque vs measured torque 10.79

Cumulative Sum Eorror [%]Modeled Torque 0.21

45

46 6 Results

2220 2225 2230 2235 2240 2245 2250 2255 2260 2265

Time [s]

100

200

300

400

Torq

ue

[Nm

]

Measured Torque

Modeled Torque

Figure 6.1: An illustration of the modelled torque together with the mea-sured torque.

6.1.2 Fuel

The result of the static and the dynamic fuel model are presented in Tables 6.2to 6.5 and Figure 6.2.The convex static fuel model has a larger relative error than the static non-convexmodel when compared to the measured values, see Table 6.2. It can be seen in Fig-ure 6.2 that the convex approximation often overestimates the static non-convexmodel and this overestimation is approximately the same in value for low valuesof fuel consumption and high values of fuel consumption, resulting in a high rel-ative error for low values of fuel consumption. This is a consequence from thatthe convex approximation is especially bad for low and negative engine torques.A consequence from this is that the convex dynamic model also performs bad forlow and negative torques.

Interesting to notice is that if the dynamic model is added to the non-convexstatic model, a much lower RMSE value is obtained, see Tables 6.2 and 6.3.

In addition, the same errors were calculated for torques above 15 newton meters(Nm) and these are presented in Table 6.4. These values show that for torquesabove 15 Nm the convex approximation is not as bad which in turn results in abetter convex dynamic model.

Table 6.2: Validation of the static fuel models.

Static Fuel Cumulative Sum Error [%]Non-Convex Model vs Measurements 0.0022Convex Model vs Measurements 5.42Convex Model vs Non-convex Model 5.23

RMSE [%]Non-convex Model vs Measurements 4.83Convex Model vs Measurements 11.00

6.1 Models 47

Table 6.3: Validation of the dynamic fuel model.

Dynamic Fuel Cumulative Sum Error [%]Convex Model vs Measurements 5.43

RMSE [%]Convex Model vs Measurement 8.58Non-convex Model vs Measurement 2.28

Table 6.4: Validation of the static fuel models, using data for engine torquesabove 15 Nm.

Static Fuel RMSE [%]Non-convex Model vs Measurements 5.75Convex Model vs Measurements 9.77

Table 6.5: Validation of the dynamic fuel model, using data for enginetorques above 15 Nm.

Dynamic Fuel RMSE [%]Convex Dynamic Model vs Measurements 5.37Non-convex Dynamic Model vs Measurements 2.66

2140 2145 2150 2155 2160 2165

Time [s]

0.5

1

1.5

2

2.5

ṁf[kg/

s]

×10-3

Static Model

Measured Fuel

Convex Static model

Convex Dynamic model

Figure 6.2: All fuel models in the same figure to illustrate how they relate toeach other. The behaviour of the fuel mass flow is a consequence of torquesteps made for the engine.

6.1.3 NOx

In this subsection, the results of the static and the dynamic NOx models are pre-sented. The difference between the convex static model and the static non-convexmodel is larger for NOx than for fuel. The convex NOx model also overestimates

48 6 Results

the NOx consumption and the approximation is worse for low and negative en-gine torques. The improvement by only considering engine torques above 15 Nmis better for NOx than for fuel.

Table 6.6: Validation of the static NOx models.

Static NOx Cumulative Sum Error[%]Non-convex Model vs Measurements 77.29Convex Model vs Measurements 77.54Non-convex Model vs Convex Model 1.15

Variance Error [%]Non-convex Model vs Measurements 0.0383Convex Model vs Measurements 0.0355

RMSE [%]Convex Model vs Non-convex Model 29.18Convex Model vs Non-convex Model (TICE above 15Nm) 18.85

Table 6.7: Validation of the dynamic NOx model.

Dynamic NOx Cumulative Sum Error [%]Measurements vs Convex Model 2.05 · 10−6

0

1

2

ṁNOx[a.u.]

ṁNOx

Measured values

Dynamic Model

3075 3076 3077 3078 3079 3080 3081 3082 3083

Time [s]

0

100

200

300

T [

Nm

]

Torque

Figure 6.3: Dynamic NOx model together with measured NOx values wherethe dynamic NOx is smaller than the measured value.

6.1 Models 49

0

2

4

6

ṁNOx[a.u.]

ṁNOx

Measured values

Dynamic Model

1715 1716 1717 1718 1719 1720 1721 1722 1723

Time [s]

-100

0

100

200

T [

Nm

]

Torque

Figure 6.4: Dynamic NOx model together with measured NOx where thedynamic NOx is higher than the measured values.

By looking at Figure 6.5, we see that the difference in NOx peaks for the modeland the measurement is very big for a torque step. However, in Figure 6.4 a sim-ilar torque step is made and the NOx model and the measurement show similarbehaviour.

0

2

4

6

ṁNOx[a.u.]

ṁNOx

Measured values

Dynamic Model

745 746 747 748 749 750 751 752 753

Time [s]

0

100

200

300

T [

Nm

]

Torque

Figure 6.5: Dynamic NOx model together with measured NOx.

50 6 Results

In Figure 6.6 the static NOx model and the convex static NOx model are illus-trated. It is seen that the convex static model often overestimates the static model.

1040 1060 1080 1100 1120 1140 1160

Time [s]

ṁNOx

Static Model

Convex Static Model

Figure 6.6: Static NOx model and Convex Static NOx model.

6.2 Optimization

Below, results obtained from the optimization is presented and commented topoint out interesting phenomena.

The engine torque trajectories obtained from the global optimization and theMPC for a set of different prediction horizons are presented in Figure 6.7. The re-sults are shown for only the WLTC drive cycle because the behaviour is the samefor each drive cycle. Only the dynamic MPC is presented due to the solution ob-tained from the static MPC, independent of prediction horizon, is similar to thesolution obtained from the static global optimization. In Figure 6.7 it is worthnoticing the difference in how the torque is requested from the ICE and the ISGwhen comparing the global static solution and the dynamic global solution. An-other interesting fact is how the optimization is able to use the the two actuatorsfor different prediction horizons. Several different prediction horizons were ana-lyzed, but to demonstrate the major characteristics the prediction horizons usedin Figure 6.7 are enough. In Figure 6.8 the resulting fuel mass flow and NOxmass flow are illustrated.

6.2 Optimization 51

Figure 6.7: A time slot of the WLTC drive cycle that visualizes how TICE,act ,TICE,req, and TISG differ if the controller has no knowledge about the dy-namic behavior (row one), has limited knowledge of the dynamics ahead(rows two and three) and when it has complete knowledge of the dynamicsfor the drive cycle (row four).

52 6 Results

Figure 6.8: A time slot of the WLTC drive cycle that visualizes how the fuelmass flow and NOx emissions differ if the controller has no knowledge aboutthe dynamic behavior (row one), has limited knowledge of the dynamicsahead (rows two and three) and when it has complete knowledge of the dy-namics for the drive cycle (row four).The figures to the left a

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